Cyclotomic polynomial

From Citizendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In algebra, a cyclotomic polynomial is a polynomial whose roots are a set of primitive roots of unity. The n-th cyclotomic polynomial, denoted by Φn has integer cofficients.

For a positive integer n, let ζ be a primitive n-th root of unity: then

The degree of is given by the Euler totient function .

Since any n-th root of unity is a primitive d-th root of unity for some factor d of n, we have

By the Möbius inversion formula we have

where μ is the Möbius function.

Examples